Four algorithms to solve symmetric multi-type non-negative matrix tri-factorization problem

Rok Hribar (), Timotej Hrga, Gregor Papa, Gašper Petelin, Janez Povh, Nataša Pržulj and Vida Vukašinović
Additional contact information
Rok Hribar: Jožef Stefan Institute
Timotej Hrga: University of Ljubljana
Gregor Papa: Jožef Stefan Institute
Gašper Petelin: Jožef Stefan Institute
Janez Povh: University of Ljubljana
Nataša Pržulj: Institute of Mathematics, Physics and Mechanics
Vida Vukašinović: Jožef Stefan Institute

Journal of Global Optimization, 2022, vol. 82, issue 2, No 4, 283-312

Abstract: Abstract In this paper, we consider the symmetric multi-type non-negative matrix tri-factorization problem (SNMTF), which attempts to factorize several symmetric non-negative matrices simultaneously. This can be considered as a generalization of the classical non-negative matrix tri-factorization problem and includes a non-convex objective function which is a multivariate sixth degree polynomial and a has convex feasibility set. It has a special importance in data science, since it serves as a mathematical model for the fusion of different data sources in data clustering. We develop four methods to solve the SNMTF. They are based on four theoretical approaches known from the literature: the fixed point method (FPM), the block-coordinate descent with projected gradient (BCD), the gradient method with exact line search (GM-ELS) and the adaptive moment estimation method (ADAM). For each of these methods we offer a software implementation: for the former two methods we use Matlab and for the latter Python with the TensorFlow library. We test these methods on three data-sets: the synthetic data-set we generated, while the others represent real-life similarities between different objects. Extensive numerical results show that with sufficient computing time all four methods perform satisfactorily and ADAM most often yields the best mean square error (MSE). However, if the computation time is limited, FPM gives the best MSE because it shows the fastest convergence at the beginning. All data-sets and codes are publicly available on our GitLab profile.

Keywords: Non-negative matrix factorization; Fixed point method; Block coordinate descent; Projected gradient method; ADAM (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10898-021-01074-3

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